The millionaire and longstanding Vladimir Putin ally said in a video published on Telegram that the handover will be completed by 1 June.
There was no immediate comment from the Russian defence ministry.
It was not possible to verify whether Wagner forces have begun to pull out of the heavily damaged city.
On Wednesday, the Ukrainian General Staff said heavy fighting was continuing inside Bakhmut after a nine-month battle, which has killed tens of thousands of people.
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Ukraine’s deputy defence minister said on Thursday that Wagner units have been replaced by regular troops in the suburbs but Wagner fighters remain in the centre of the city.
Ukrainian forces still have a foothold in the southwestern outskirts, Hanna Maliar said.
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Image: Mr Prigozhin with two of his mercenaries ‘Biber’ and ‘Dolik’
A Wagner victory in Bakhmut would be a much-needed boost for Mr Putin after his invasion lost momentum.
Top Ukrainian presidential adviser Mykhailo Podolyak said Ukraine’s long-awaited counteroffensive is already under way, saying it should not be anticipated as a “single event” starting “at a specific hour of a specific day”.
He said on Twitter that “dozens of different actions to destroy Russian occupation forces” were “taking place yesterday, are taking place today and will continue tomorrow”.
Mr Prigozhin has been in a long-running feud with Moscow and, during the 15-month war, he has repeatedly chastised Russia’s military leadership, accusing them of incompetence and failure to properly supply his troops.
Image: Mr Prigozhin has been critical Russia’s tactics in Bakhmut
Wagner’s involvement in the capture of Bakhmut has upped Mr Prigozhin’s standing, which he has used to set out his personal views about the war.
“Prigozhin is … using the perception that Wagner is responsible for the capture of Bakhmut to advocate for a preposterous level of influence over the Russian war effort in Ukraine,” according to the Institute for the Study of War.
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3:46
Analysis: Wagner boss appears to insult Putin
His frequent critical commentary about Russia’s military performance is unusual under Mr Putin’s censorious regime.
His latest statement on Bakhmut came a day after he again broke with the Kremlin line on Ukraine.
He said its goal of demilitarising the country has backfired, acknowledged Russian troops have killed civilians and agreed with western estimates that he lost more than 20,000 men in the battle for the city.
Regime change in Iran is “unacceptable” and the assassination of the country’s supreme leader would “open the Pandora’s box”, the Kremlin has said.
In a rare interview with a foreign media organisation, Kremlin spokesman Dmitry Peskov told Sky News that Russia would react “very negatively” if Ayatollah Ali Khamenei was killed.
The comments came as US President Donald Trump said he will decide within two weeks whether America will join Israel’s military campaign against Tehran, after earlier speculating on social media about killing the Iranian leader.
Image: Dmitry Peskov speaks to Sky News
“The situation is extremely tense and is dangerous not only for the region but globally,” Mr Peskov said in an interview at the Constantine Palace in Saint Petersburg.
“An enlargement of the composition of the participants of the conflict is potentially even more dangerous.
“It will lead only to another circle of confrontation and escalation of tension in the region.”
Image: Putin and Khamenei meeting in Tehran in 2022. Pic: AP
They are the Kremlin’s strongest comments yet regarding the Israel-Iran conflict, which has stoked fears in Moscow that it could be on the verge of losing its closest ally in the Middle East.
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Russia has deepened its ties with Iran since invading Ukraine, and the two countries signed a strategic partnership in January.
“[Regime change in Iran] is unimaginable. It should be unacceptable, even talking about that should be unacceptable for everyone,” Mr Peskov said, in a thinly veiled reference to Washington.
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1:35
How will Russia react to US joining Israel?
But Mr Peskov refused to be drawn on what action Russia would take if Khamenei was killed, saying instead it would trigger action “from inside Iran”.
“It would lead to the birth of extremist moods inside Iran and those who are speaking about [killing Khamenei], they should keep it in mind. They will open the Pandora’s box.”
Vladimir Putin’s offers to mediate an end to the conflict have so far been rejected by Mr Trump, who said on Wednesday that he told the Russian president to “mediate your own [conflict]”, in reference to Russia’s war against Ukraine.
Mr Peskov denied the American president’s words were insulting, adding: “Everyone has a different language.
“President Trump has his own unique way of speaking and his unique language. We are quite tolerant and expect everyone to be tolerant of us.”
Image: Trump’s attempts to broker peace between Ukraine and Russia have so far not been fruitful. Pic: AP
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The Trump administration’s own mediation efforts to end the war in Ukraine have failed to yield any major breakthroughs, despite two rounds of direct talks between Moscow and Kyiv.
Moscow has stepped up its aerial bombardment of Ukraine in recent weeks and continues to reject Volodymyr Zelenskyy’s calls for a 30-day ceasefire.
“Now we have a strategic advantage. Why should we lose it? We are not going to lose it. We are going further. We’re advancing and we’ll continue to advance,” Mr Peskov said.
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Russia has previously said it would only commit to a ceasefire if Kyiv stops receiving foreign military support, fearing that a pause in the fighting would offer Ukraine a chance to rearm and regroup its forces.
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0:57
Russia ‘relentlessly terrorises’ Kyiv, says Zelenskyy
Asked if Moscow could commit to not using a ceasefire in the same way, Mr Peskov said: “A ceasefire is a ceasefire, and you stop.
“But America is not saying that ‘we’ll quit any supplies’. Britain is not saying that as well. France is not saying that as well. This is the problem.”
China has criticised a British warship’s passage through the Taiwan Strait as a deliberate move to “cause trouble”.
The Royal Navy said its patrol vessel HMS Spey was conducting a routine navigation through the contested waterway on Wednesday as part of a long-planned deployment in compliance with international law.
In response, the Eastern Theatre Command of China’s People’s Liberation Army (PLA) said the exercise was “public hyping”, adding that its forces followed and monitored the ship.
“The British side’s remarks distort legal principles and mislead the public; its actions deliberately cause trouble and disrupt things, undermining peace and stability in the Taiwan Strait,” it said in a statement on Friday.
“Troops in the theatre are on high alert at all times and will resolutely counter all threats and provocations.”
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3:23
From 2024: Why is South China sea so disputed?
The strait is contested between Taiwan and China, which split in 1949.
Today, China views Taiwan as a breakaway province – with which it promises to one day reunify, and has not ruled out the use of force to do so – and regards the waterway as its own territory.
Taiwan, the US, and other Western powers regard the strait as international waters.
US navy ships sail through the strait around once every two months, sometimes accompanied by those of allied nations.
Responding to HMS Spey’s exercise, Taiwan’s foreign ministry said it “welcomes and affirms the British side once again taking concrete actions to defend the freedom of navigation in the Taiwan Strait”.
China has also carried out several military drills across the waterway, with exercises in October involving its army, navy and rocket forces. Beijing called it a “stern warning to the separatist acts of Taiwan independence forces” at the time.
It comes as Taiwanese President Lai Ching-te on Thursday ordered defence and security units to step up their monitoring and intelligence efforts in response to China’s military activities.
Taiwan’s defence ministry also reported another spike in Chinese movements close to the island over the previous 24 hours, involving 50 aircraft, concentrated in the strait and the top part of the South China Sea.
This is the most significant statement from the US president in days, though it still keeps everyone guessing.
In a message conveyed through his press secretary, he is giving diplomacy up to two weeks to work.
“Based on the fact that there’s a substantial chance of negotiations that may or may not take place with Iran in the near future, I will make my decision whether or not to go within the next two weeks,” Karoline Leavitt quoted him as saying.
It is not clear what “whether or not to go” entails.
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0:40
Trump: Iran ‘weeks away’ from nuclear weapon
We know that he has been given a spectrum of different military options by his generals and we know that the Israelis are pressuring him to use American B2 bombers with their bunker-busting bombs to destroy Iran’s nuclear facility at Fodow.
The Israelis are encouraging no delay. But against that, he is weighing up many risks, both military and political.
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Militarily, it is not clear how successful a bunker-busting strike on Fordow would be.
Experts have suggested it would require several of the massive bombs, which have never been used in combat before, to be dropped on the site.
It is not as simple as one clean strike and job done.
Politically, the president is under significant pressure domestically not to get involved in Iran.
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2:40
MAGA civil war breaks out over Iran
Within his own MAGA coalition – influencers, politicians and media personalities are lining up in criticism of involvement in the conflict.
One of those leading the criticism, his former chief strategist Steve Bannon, who maintains huge influence, was seen entering the White House on Thursday.
His press secretary reiterated to us that the president always wants to give diplomacy a chance and she confirmed that his Middle East envoy Steve Witkoff has spoken to the Iranian foreign minister Abbas Araghchi.
Image: Steve Bannon, seen recently at a conservative event in Maryland, is against US involvement in Iran. Pic: AP
European leaders, including the UK foreign secretary David Lammy, who is in Washington, are meeting Mr Araghchi in Geneva on Friday.
The two-week window – assuming it lasts that long – also gives space to better prepare for any strike and mitigate against some of the other risks of US involvement.
There are 40,000 troops in bases across the Middle East. It takes time to increase security at these bases or to move non-essential personnel out. It also takes time to move strategic military assets into the region.
The USS Nimitz aircraft carrier and its support vessels were redeployed from the Indo-Pacific on Monday. Their last known position was the Strait of Malacca two days ago.
The Nimitz Carrier Group will overlap with the USS Carl Vinson group which was deployed to the Middle East in March.
The potential two-week window also allows for more time for a ‘day after’ plan, given that the Israeli strategy appears to be regime change from within.
Since the Israeli action in Iran began last week, the worst-case scenario of mass casualties in Israel from Iranian attacks has not materialised.
The president is said to be surprised and encouraged by this. “Israel has exceeded a lot of people’s expectations in their abilities,” press secretary Karoline Leavitt said.
The Israeli success, the absence of a mass casualty event in Israel, and the lack of any sustained counterattack by Iranian proxies in the region remove reservations that previous presidents have had about taking on Iran.
That said, sources have told Sky News that the president is determined that the diplomatic solution should be given a chance despite current pessimism over the chances of success.